A walk on the beach, in the hills or along a river bank provides great opportunities for mathematical reflection. How high is the mountain? How many grains of sand are on the beach? How much water is flowing in the river? [TM156 or search for “thatsmaths” at irishtimes.com].

While the exact answers may be elusive, we can make reasonable guesstimates using basic knowledge and simple mathematical reasoning. And we will be walking in the footsteps of some of the world’s greatest thinkers.

**The
Sand Reckoner**

Estimating the number of grains of sand on a beach occupied **Archimedes**. Indeed, he considered an even greater problem: to determine the number of grains of sand that would fill the Universe. To answer this, he had to estimate the size of the Universe and to devise a method of dealing with extremely large numbers. The largest number then in use was a *myriad* or ten thousand. Archimedes used powers of a myriad myriads to form numbers of enormous size and, in the process, discovered the rules for manipulating powers of ten.

Using
the astronomical knowledge of the day, Archimedes deduced that the
diameter of the Universe was (in modern units) about 2 light years.
He calculated the number of grains of sand to fill it, arriving at a
number 10^{63},
that is, 1 followed by 63 zeros. He summarized his musings in *The
Sand Reckoner*,
a work that has been described as the first expository mathematical
paper.

School trigonometry enables us to measure the height of a tree, the width of a river or the altitude of a mountain. Similar reasoning allowed the medieval Persian scholar **Abu Rayhan Al-Biruni** to estimate the size of the Earth using the distance from a mountain-top to the horizon. The value obtained by Al-Biruni for the Earth’s circumference was in good agreement with the value of the circumference of the Earth known today.

**Shannon Flow**

To estimate the volume of water flowing in a river, we could guess the cross-section area and flow speed and work from there, or we could use the annual rainfall to obtain the mean flow. Let’s consider the River Shannon: Ireland is about 200km (east-west) by 400km (north-south) giving an area of 80,000 square kilometres. If the Shannon basin area is one fifth of this, we get 16,000 square kilometres.

To estimate the volume of water flowing in a river, we could guess the cross-section area and flow speed and work from there, or we could use the annual rainfall to obtain the mean flow. Let’s consider the River Shannon: Ireland is about 200km (east-west) by 400km (north-south) giving an area of 80,000 square kilometres. If the Shannon basin area is one fifth of this, we get 16,000 square kilometres.

The annual rainfall is about 1 metre so, if half the rainfall evaporates, the volume of water flowing to the sea is 8 billion cubic metres. There are about 32 million seconds in a year, so the mean outflow is something like 250 cubic metres every second. The ESB measures the daily average flow rate at Ardnacrusha near Limerick (see figure at top of this post). The flow varies throughout the year, peaking at about 400 cubic metres per second, so the above estimate is reasonable.

**Challenges
Everywhere**

Numerous problems like this can enrich a ramble. How fast do circular waves radiate outwards from a stone thrown in a pond? How do the criss-cross patterns form when the waves interact? Are rainbows always circular and always the same size? How fast do birds fly? Does a spider’s web have a simple mathematical description? What determines the length-scale of the sand ridges on a flat tidal beach? By how much might the visibility deteriorate if heavy rain changes to drizzle, assuming droplet-size decreases by a factor of ten and the same amount of water is falling?

Some of the wonders encountered on a ramble can easily be explained, while others continue to challenge leading scholars. But the fun is in the effort, and real joy comes when a sudden flash of insight occurs.